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Mirrors > Home > MPE Home > Th. List > rankdmr1 | Structured version Visualization version GIF version |
Description: A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankdmr1 | ⊢ (rank‘𝐴) ∈ dom 𝑅1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankidb 9229 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
2 | elfvdm 6702 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → suc (rank‘𝐴) ∈ dom 𝑅1) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) ∈ dom 𝑅1) |
4 | r1funlim 9195 | . . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
5 | 4 | simpri 488 | . . . 4 ⊢ Lim dom 𝑅1 |
6 | limsuc 7564 | . . . 4 ⊢ (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1) |
8 | 3, 7 | sylibr 236 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
9 | rankvaln 9228 | . . 3 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅) | |
10 | limomss 7585 | . . . . 5 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
11 | 5, 10 | ax-mp 5 | . . . 4 ⊢ ω ⊆ dom 𝑅1 |
12 | peano1 7601 | . . . 4 ⊢ ∅ ∈ ω | |
13 | 11, 12 | sselii 3964 | . . 3 ⊢ ∅ ∈ dom 𝑅1 |
14 | 9, 13 | eqeltrdi 2921 | . 2 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ∈ dom 𝑅1) |
15 | 8, 14 | pm2.61i 184 | 1 ⊢ (rank‘𝐴) ∈ dom 𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 dom cdm 5555 “ cima 5558 Oncon0 6191 Lim wlim 6192 suc csuc 6193 Fun wfun 6349 ‘cfv 6355 ωcom 7580 𝑅1cr1 9191 rankcrnk 9192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-r1 9193 df-rank 9194 |
This theorem is referenced by: r1rankidb 9233 pwwf 9236 unwf 9239 uniwf 9248 rankr1c 9250 rankelb 9253 rankval3b 9255 rankonid 9258 rankssb 9277 rankr1id 9291 |
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